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Nonparametric Shape-Restricted Regression
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DOI：10.1214/18-STS665.png)
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摘要
En 中文
We consider the problem of nonparametric regression under shape constraints. The main examples include isotonic regression (with respect to any partial order), unimodal/convex regression, additive shape-restricted regression and constrained single index model. We review some of the theoretical properties of the least squares estimator (LSE) in these problems, emphasizing on the adaptive nature of the LSE. In particular, we study the behavior of the risk of the LSE, and its pointwise limiting distribution theory, with special emphasis to isotonic regression. We survey various methods for constructing pointwise confidence intervals around these shape-restricted functions. We also briefly discuss the computation of the LSE and indicate some open research problems and future directions.
Keyword:
Adaptive risk bounds
bootstrap
Chernoff's distribution
convex regression
isotonic regression
likelihood ratio test
monotone function
order preserving function estimation
projection on a closed convex set
tangent cone
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